**Nomenclature**

*Ta,m* ( ◦C) Ambient temperature when measuring the *m*-th piece. *Ta,*<sup>0</sup> ( ◦C) Ambient temperature when measuring the master piece. *Ts,m* ( ◦C) System temperature when measuring the *m*-th piece. *Ts,*<sup>0</sup> ( ◦C) System temperature when measuring the master piece. *Tm,m* ( ◦C) Measurand temperature when measuring the *m*-th piece. *Tm,*<sup>0</sup> ( ◦C) Measurand temperature when measuring the master piece. *Tc* ( ◦C) Temperature from the contact thermometer (used in the Monte Carlo simulation). *Ta* ( ◦C) Temperature from the ambient thermometer (used in the Monte Carlo simulation). *Hs,m* (mm) Tooling height of the system (see Figure 2) when measuring the *m*-th piece. *Hs,*<sup>0</sup> (mm) Tooling height of the system (see Figure 2) when measuring the master piece. *H* (mm) Tooling height of the system at 20 ◦C. *Lm,m* (mm) Probe length when measuring the *m*-th piece. *L*0*,m* (mm) Probe length from the measurement of the master piece but when measuring the *m*-th piece. *L*0*,*<sup>0</sup> (mm) Probe length when measuring the master piece. *Lm* (mm) Probe length when measuring the *m*-th piece (dimension at 20 ◦C). *L*<sup>0</sup> (mm) Probe length when measuring the master piece (dimension at 20 ◦C). *Dm,m* (mm) Measurand (diameter) of the *m*-th piece at the moment of measure it. *D*0*,m* (mm) Measurand (diameter) of the master piece at the moment of measure the *m*-th piece. *D*0*,*<sup>0</sup> (mm) Measurand (diameter) of the master piece at the moment of measure it. *Dm* (mm) Measurand (diameter) of the *m*-th piece (dimension at 20 ◦C). *D*<sup>0</sup> (mm) Measurand (diameter) of the master piece (dimension at 20 ◦C). Δ*Lm* (mm) Figure, at 20 ◦C, of the difference between *L*0,*<sup>m</sup>* and *Lm,m*, see Equations (8) and (9). *θm* ( ◦) Angle between a vertical line and the front face of the ring (see Figure 4) when measuring the *m*-th piece. *θ*<sup>0</sup> ( ◦) Angle between a vertical line and the front face of the ring (see Figure 4) when measuring the master piece. *ψm* ( ◦) Angle between a vertical line and the axis of the contact prober of the comparator dial when measuring the *m*-th piece (see Figure 5). *ψ*<sup>0</sup> ( ◦) Angle between a vertical line and the axis of the contact prober of the comparator dial when measuring the master piece (see Figure 5). *λm* ( ◦) Linear translation on X-axis direction (see Figure 5) when measuring the *m*-th piece. *λ*<sup>0</sup> ( ◦) Linear translation on X-axis direction (see Figure 5) when measuring the master piece. *αm* ( ◦C<sup>−</sup>1) Thermal expansion coefficient of the *m*-th piece. *α*<sup>0</sup> ( ◦C<sup>−</sup>1) Thermal expansion coefficient of the master piece.

#### *αs* ( ◦C<sup>−</sup>1) Thermal expansion coefficient of the system.

#### **References**


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